return to main page
ATTENTION !  May be best viewed by INTERNET EXPLORER  5.0 or HIGHER
THE  THEOREMS  OF  EL'EVVEL STATES THAT ;

Binomial coefficient  C(x,y)  is given by ---    

PART-1
If  p  is  an  odd  prime  and  k , m  are  any integers then we'll  have the
following  congruence relation.
 

 

such that    (k , p) = 1


This  theorem  sieves  off  many  of  the  CARMICHAEL  NUMBERS.
As a matter of fact, perhaps these kind of theorems  does't contain the concept of CARMICHAEL type of numbers, when you encounter with a pseudoprime to a certain  k,m ;  just change one or both  of   them .
Then the congruence holds true . Now  that  means  there  is  no  pseudoprime  to  all  bases   concept.
Please click here to see the related Mathematica program .


 PART-2
If  p  is  an  odd  prime  and  m , k , x are any integers then we'll  have the
following  congruence relation.
 

 

such that  (m+k , p) = 1

For x=1 ; the formula turns Fermat's Little Theorem as [a p º a mod p] Where a=m+k

 This  theorem  sieves  off  many  of  the  CARMICHAEL  NUMBERS.
As a matter of fact, perhaps these kind of theorems  does't contain the concept of CARMICHAEL type of numbers, when you encounter with a pseudoprime to a certain  m , k , x ; just change one  or  all  of   them .
Then the congruence holds true . Now  that  means  there  is  no  pseudoprime  to  all  bases   concept.
Please click here to see the related Mathematica program .


 

return to main page